By Cauchy Schwarz inequality $$(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}) \geq 9$$which desired $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 9$ Since $$\lfloor \frac{1}{a} \rfloor > \frac{1}{a} - 1$$Then $n > \frac{1}{a}+\frac{1}{b}+\frac{1}{c} - 3 \geq 6 $ Hence $n \geq 7$ is interesting ( sorry if I’m wrong. )

demmy wrote: By Cauchy Schwarz inequality $$(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}) \geq 9$$which desired $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 9$ Since $$\lfloor \frac{1}{a} \rfloor > \frac{1}{a} - 1$$Then $n > \frac{1}{a}+\frac{1}{b}+\frac{1}{c} - 3 \geq 6 $ Hence $n \geq 7$ is interesting ( sorry if I’m wrong. ) examples are missing for each numbers $n$

DanDumitrescu wrote: We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \]where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.) By CS(not really sure), $n \ge 7$. plugging $a=\frac{1}{n-4},b=c=\frac{n-5}{2n-8}$ gives us the desired result.

i cant really find examples for 7...

Try a=4/13, b=c=4.5/13, for n=7

Sunshine132 wrote: Try a=4/13, b=c=4.5/13, for n=7 i think something random ranging from 1/3 and 1/4 should work ig like $\frac{2}{7},\frac{5}{14},\frac{5}{14}$ but thank you nonetheless

Jjesus wrote: demmy wrote: By Cauchy Schwarz inequality $$(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}) \geq 9$$which desired $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 9$ Since $$\lfloor \frac{1}{a} \rfloor > \frac{1}{a} - 1$$Then $n > \frac{1}{a}+\frac{1}{b}+\frac{1}{c} - 3 \geq 6 $ Hence $n \geq 7$ is interesting ( sorry if I’m wrong. ) examples are missing for each numbers $n$ I forgot T-T. Thanks for telling me . Just construct like @melowmolly